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SEPARATION of the «-SPHERE by an (N - 1)-SPHERE by JAMES C

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SEPARATION of the «-SPHERE by an (N - 1)-SPHERE by JAMES C SEPARATION OF THE «-SPHERE BY AN (n - 1)-SPHERE BY JAMES C. CANTRELL(i) 1. Introduction. Let A be the closed spherical ball in E" centered at the origin 0, and with radius one, B the closed ball centered at O with radius one-half, and C the closed ball centered at O with radius two. The Generalized Schoenflies Theorem states that, if h is a homeomorphism of Cl (C —B) into S", then h(BdA) is tame in S" (the closure of either component of S" —h(BdA) is a closed n-cell) [5]. One is naturally led to the following question : if h is a homeomorphism of C\(A—B) into S", is the closure of the component of S" — h(BdA) which contains n(BdB) a closed n-cell? This question is answered affirmatively by Theorem 1 and should be listed as a corollary to the Generalized Schoenflies Theorem. Let D be the closed ball in E", centered at (0,0,-",0, — 1) with radius two. Two other types of embeddings of Bd.4 in S", n > 3, are considered in §2, (1) the embedding homeomorphism h can be extended to a homeomorphism of Cl(£>— B) into S" such that the extension is semi-linear on each finite polyhedron in the open annulus Int (A — B), and (2) h can be extended to a homeomorphism of C1(B —A) into S" such that the extension is semi-linear in a deleted neigh- borhood of (0, 0, ■■■,(), 1) (see Definition 1). Theorem 4 strongly suggests that, for an embedding of type (1), h(BdA) is tame in S". An embedding of this type corresponds to the three dimensional case in which h(BdA) is locally polyhedral except at one point. In §3, three methods of constructing 3-spheres in S4 from 2-spheres in S3 are considered: (1) suspension of a 2-sphere in S3, (2) rotation of a 2-cell in S3 about the plane of its boundary, and (3) capping a cylinder over a 2-sphere in S3. The construction methods in cases (1) and (2) were introduced by Artin [2] and have been used by him and by Andrews and Curtis [1] to construct 2-spheres in S4 from 1-spheres in S3. Their techniques may be applied directly to establish isomorphism theorems relating the fundamental groups of the complements of the constructed 3-spheres and the fundamental groups of the corresponding complements of the given 2-spheres. Thus, methods (1) and (2) may be used to construct wild (nontame) 3-spheres in S4. Method (2) is also used to construct Presented to the Society, November 17,1961 under the title Some embeddings ofS"-1 in S"; received by the editors July 24, 1962. 0) The author wishes to express his appreciation to Professor O. G. Harrold, Jr., for directing the thesis, the principal results of which constitute this paper. This research was supported in part through the NSF-G8239. 185 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 186 J. C. CANTRELL [August a 3-sphere in S4, one complementary domain of which is simply connected but is not an open 4-cell. The third method is used to construct a 3-sphere in S4 such that one complementary domain has a closure which is a closed 4-cell, and the other complementary domain is an open 4-cell but its closure is not a closed 4-cell. 2. Some embeddings of S"_1inS". The reader is referred to [5] for the definitions of inverse set and cellular set. Theorem 1. Let h be a homeomorphism ofC\(A - B) into S" and let G be the component of S" — h(BdA) which contains h(BdB). Then C1G is a closed n-cell Proof. Let G' be the component of S" - h(BdB) which does not contain h(BdA). We first observe that C1G' is a cellular subset of G. For, if B¡ is the closed ballin£", centered at O with radius 1/2 + l/(i+2), i = l,2,---,and Gt is the component of Sn— /t(BdB¡) which contains G', then, by the Generalized Schoen- flies Theorem, C1G¡ is a closed n-cell. Furthermore ClGi+1 c G. and p|7=1ClG, = ClG'. Let g be a continuous mapping of C1L4 — B) onto A such that Bdß is the only inverse set. Define a mapping/ of C1G onto A by the equations f(x) = gh~\x), if xeClG-G', f(x) = g(BdB), if xeG'. The mapping / carries C1G continuously onto A such that the only inverse set is the cellular subset C1G' of G. Thus, by Theorem 2 of [5], C1Gis a closed n-cell. Theorem 2. Let h be a homeomorphism ofCl(D — B) into S" and G be let the component of S" — h(BdA) which intersects h(BdD). Then G is an open n-cell. Proof. Let H be the component of S" - h(BdA) which contains h(BdB). By Theorem 1, CIH is a closed n-cell and, hence, there is a homeomorphism/ of A onto Cl// such that/ and n agree on BdA. Define a homeomorphism <J>of D into S" by the equations <j)(x)= h(x), if xeD —A, <p(x)=f(x), if xeA. Let 0[(O,O, --^O, 1)] = p and let g be a continuous mapping of D onto D such that, (1) g is fixed on BdD, (2) g is a homeomorphism of D - A onto D - (0,0,- •-,0,1), and (3) g(A) = (0,0,-",0,1). Now define a continuous mapping \¡i of S" onto S" by the equations \¡/(x) = x, if xe S"-<t>(D), i¡/(x)= <l>g<l>-1(x),if xe 0(D). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1963] SEPARATION OF THE «-SPHERE BY AN («-1)-SPHERE 187 The mapping \¡j carries S" onto S", leaves p fixed, and has CIH as the only inverse set. Hence, G is carried homeomorphically onto S" — p, and is an open n-cell. Let By be the closed ball in E" which is centered at O and has radius three- fourths, and let L' be the closed segment of the x„-axis from (0,0, •••,0,3/4) to (0,0, -.0,1). Theorem 3. Let h be a homeomorphism ofC\(D — B) into S" and denote h(L') by Land «(0,0, "-,0,1) by p. Let G be the component of S"—h(BdA) which intersects n(BdD) and let H be the component of S" — h(BdB¡) which contains h(BdA). Then C\H is a closed n-cell and (C1G) —p is topologically equivalent to CIH - L. Proof. That CIH is a closed n-cell follows immediately from Theorem 1. Let K be the component of S" — h(BdD) which does not intersect h(BdA) and let g be a continuous mapping of C1(D — By) onto C1(D — A) such that (1) g is fixed on BdD, (2) g(BdBy) = BdA, and (3) L is the only inverse set under g. The mapping / of CIH onto C1G defined by f(x) = x, if Ex e K,ï f(x) = hgh~\x), if x eClH - K, is a continuous mapping of CIH onto C1G such that the only inverse set is Land f(L) = p. Hence, / is a homeomorphism of CIH - L onto C1G — p. If in Theorem 3 there exists a continuous mapping k of CIH onto CIH such that L is the only inverse set, then we can state that C1G is a closed n-cell. In fact, the product mapping kf1 is a homeomorphism of C1G onto CIH. Let us now suppose that n > 3 and that h is semi-linear on each finite polyhedron of Int (A —B) (we assume a curved decomposition of E" in which A, B, By, and L are polyhedra). Then h(BdBy) is a polyhedron and L is locally polyhedral except at p. Let s > 0 be such that S(s,p) c H and use Lemma 2 of [6] to obtain a homeomorphism <pof S" onto S" such that c¡>is fixed outside S(e,p) and <p(L) is polyhedral. Let q be the endpoint of L which lies on BdH and let g be a polyhedral n-cell in CIH such that q e BdQ, 4>(L)— q cz lnt Q, and Q has a subdivision isomorphic to a subdivision of a simplex (see [7, Lemma 5.3]). Let ^ be a semi- linear homeomorphism of Q onto a simplex R. The arc ^i(p(L) is then polyhedral in R and, together with the linear segment \j/<p(q)\l/(p(p),from \¡J(¡>(q)to i¡/(p(p), bounds a polyhedral 2-cell which, except for \¡/(h(q), lies in the interior of R. Lemma 3 of [9] is then applied to obtain a homeomorphism n of R onto R such that n is fixed on BdB and carries \¡/<p(L)onto \¡j<fr(q)\l/<¡>(p).It is then easy to find a continuous mapping 0 of R onto R such that 6 is fixed on BdB, 9(i¡/(j)(q)il/(j)(p)) = \jj(t>(q),and i]/(p(q)ij/(p(p)h the only inverse set. The mapping k, defined by k(x) = <p(x), ifx^tß), k(x) = ib-10#<Kx), if x e 4>-\Q), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 188 J.
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